#8429354

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4473 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(9,10,14,18)(11,12,16,15)(13,17,22,21)(19,20,24,23)(25,26)(27,31)(28,36)(29,33)(30,34)(32,37)(35,38)(39,40);(1,2)(3,8)(4,6)(5,7)(9,11,14,16)(10,15,18,12)(13,19,22,24)(17,23,21,20)(27,34)(30,31)(35,40)(38,39);(1,3,5,4)(2,6,7,8)(25,27)(26,30)(28,35)(29,34)(31,33)(32,38)(36,39)(37,40);(1,2)(3,6)(4,8)(5,7)(9,10)(11,15)(12,16)(13,21)(14,18)(17,22)(19,20)(23,24)(25,27)(26,31)(28,35)(29,34)(30,33)(32,39)(36,38)(37,40);(9,12,14,15)(10,16,18,11)(13,20,22,23)(17,24,21,19)(25,28)(26,32)(27,35)(29,37)(30,38)(31,39)(33,36)(34,40);(1,4,5,3)(2,6,7,8)(9,13)(10,17)(11,19)(12,20)(14,22)(15,23)(16,24)(18,21)(25,26)(27,30)(28,32)(29,33)(31,34)(35,38)(36,37)(39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8336550);